Our models violate the parity symmetry, which is clear from the S-matrix. However, the extended parity which interchanges
the two components is preserved (see [1]}).
As noted in [2]}, the parity symmetry is essential for the Bisognano-Wichmann
property in two dimensions. Although this is not necessarily related to our models, we present
a simple counterexample. One takes the complex free field, which admits \(S^1\) -inner symmetry
with Bisognano-Wichmann property. Then one can simply replace the Lorentz boosts by
the composition of Lorentz boosts and the inner symmetry. This still satisfies all the axioms of
net but violates the parity symmetry which must have the appropriate commutation relation with
the boosts.
Accordingly, the inner symmetry and Poincaré symmetry do not necessarily commute.
For example, we can take a net with a noncommutative Lie group symmetry and replace the
boosts as above.
The proof in four dimensions [3]} does not work in two dimensions
since the Lorentz group is abelian, and hence the Poincaré group has finite dimensional
unitary representations.
